arXiv subjects

Marcus C. Werner

Publications and source records attributed to Marcus C. Werner.

17 recordsLinked to original sources

arXiv2024

Klein Model for Hyperbolic Neural Networks

Hyperbolic neural networks (HNNs) have been proved effective in modeling complex data structures. However, previous works mainly focused on the Poincaré ball model and the hyperboloid model as coordinate representations of the hyperbolic space, often neglecting the Klein model. Despite this, the Klein model offers its distinct advantages thanks to its straight-line geodesics, which facilitates the well-known Einstein midpoint construction, previously leveraged to accompany HNNs in other models. In this work, we introduce a framework for hyperbolic neural networks based on the Klein model. We provide detailed formulation for representing useful operations using the Klein model. We further study the Klein linear layer and prove that the "tangent space construction" of the scalar multiplication and parallel transport are exactly the Einstein scalar multiplication and the Einstein addition, analogous to the Möbius operations used in the Poincaré ball model. We show numerically that the Klein HNN performs on par with the Poincaré ball model, providing a third option for HNN that works as a building block for more complicated architectures.

cs.LG↗

arXiv2022

Finsler pp-waves and the Penrose Limit

The Penrose plane wave limit is a remarkable property of Lorentzian spacetimes. Here, we discuss its extension to Finsler spacetimes by introducing suitable lightlike coordinates and adapting the Lorentzian definition of pp-waves. New examples of such Finsler pp-waves are also presented.

math.DG↗

arXiv2019

The Struble-Einstein Correspondence

This article presents hitherto unpublished correspondence of Struble in 1947 with Menger, Chandrasekhar, and eventually Einstein, about a possible observational test supporting Einstein's special relativity against Ritz's emission theory using binary stars. This `Struble effect,' an acceleration Doppler effect in emission theory, appears to have been overlooked, and the historical context, including de Sitter's binary star test of special relativity, is also discussed.

gr-qc↗

arXiv2019

Gravitational lensing in area metric spacetimes

We consider light propagation as a probe of non-metricity in area metric spacetimes, and find a deviation from the standard Etherington relation for linearized area metric Schwarzschild. This is joint work with Frederic P. Schuller (Erlangen University).

gr-qc↗

arXiv2019

Jacobi-Maupertuis Randers-Finsler metric for curved spaces and the gravitational magnetoelectric effect

In this paper we return to the subject of Jacobi metrics for timelike and null geodsics in stationary spactimes, correcting some previous misconceptions. We show that not only null geodesics, but also timelike geodesics are governed by a Jacobi-Maupertuis type variational principle and a Randers-Finsler metric for which we give explicit formulae. The cases of the Taub-NUT and Kerr spacetimes are discussed in detail. Finally we show how our Jacobi-Maupertuis Randers-Finsler metric may be expressed in terms of the effective medium describing the behaviour of Maxwell's equations in the curved spacetime. In particular, we see in very concrete terms how the magnetolectric susceptibility enters the Jacobi-Maupertuis-Randers-Finsler function.

gr-qc↗

arXiv2019

The gravitational magnetoelectric effect

Electromagnetism in spacetime can be treated in terms of an analogue linear dielectric medium. In this paper, we discuss the gravitational analogue of the linear magnetoelectric effect, which can be found in multiferroic materials. While this is known to occur for metrics with non-zero mixed components, we show how it depends on the choice of spatial formalism for the electromagnetic fields, including in differences in tensor weight, and also on the choice of coordinate chart. This is illustrated for Langevin-Minkowski, four charts of Schwarzschild spacetime, and two charts of pp gravitational waves.

gr-qc↗

arXiv2019

The Isoperimetric Problem in Riemannian Optical Geometry

In general relativity, spatial light rays of static spherically symmetric spacetimes are geodesics of surfaces in Riemannian optical geometry. In this paper, we apply results on the isoperimetric problem to show that length-minimizing curves subject to an area constraint are circles, and discuss implications for the photon spheres of Schwarzschild, Reissner-Nordstrom, as well as continuous mass models solving the Tolman-Oppenheimer-Volkoff equation. Moreover, we derive an isoperimetric inequality for gravitational lensing in Riemannian optical geometry, using curve shortening flow and the Gauss-Bonnet theorem.

gr-qc↗

arXiv2017

Etherington's Distance Duality with Birefringence

We consider light propagation in a spacetime whose kinematics allow weak birefringence, and whose dynamics have recently been derived by gravitational closure. Revisiting the definitions of luminosity and angular diameter distances in this setting, we present a modification of the Etherington distance duality relation in a weak gravitational field around a point mass. This provides the first concrete example of how the non-metricities implied by gravitational closure of birefringent electrodynamics affect observationally testable relations.

gr-qc↗

arXiv2017

Light Deflection by a Rotating Global Monopole Spacetime

We investigate the deflection of light by a rotating global monopole spacetime and a rotating Letelier spacetime in the weak deflection approximation. To this end, we apply the Gauss-Bonnet theorem to the corresponding osculating optical geometries and show that the deflection of light increases due to the presence of the global monopole parameter and the string cloud parameter, respectively. The results obtained for the deflection angle in the equatorial plane generalize known results for the corresponding non-rotating global monopole and Letelier spacetimes as well as the Kerr solution.

gr-qc↗

arXiv2016

Modifications to the Etherington Distance Duality Relation and Observational Limits

The Etherington distance duality relation, which relates the luminosity distance, the angular diameter distance and the redshift of objects, depends only upon a conservation law for light that traces back directly to the Lorentzian spacetime geometry. We show that this duality relation indeed survives transition to the most general linear electrodynamics without birefringence, which rests on a spacetime geometry constituted by a Lorentzian metric and two scalar fields, a dilaton and an axion. By computing the Poynting vector and optical scalar transport in the geometrical optics limit of this framework, we derive the modification of the light flux in the presence of a dilaton field and present improved constraints on the gradient of the dilaton field from observations of the Cosmic Microwave Background spectrum. Although this flux modification may seem applicable also to fundamental modifications of the Etherington relation, we show that the distance duality relation still holds true. Thus any deviations within this classical theory would imply non-metricities, once astrophysical sources for attenuation, such as dust, are accounted for. Moreover, using the most up-to-date measurements of the luminosity distances to Supernovae of Type Ia, and the inferred angular diameter distances from the baryon acoustic feature measurements, we perform a scale-free and nearly model-independent test of the Etherington distance duality relation between redshifts of $0.38$ and $0.61$. We find consistency with the standard distance duality relation and constrain the optical depth of light between these two redshifts to $Δτ=-0.006\pm0.046$.

astro-ph.CO↗

arXiv2016

The geometry of gravitational lensing magnification

We present a definition of unsigned magnification in gravitational lensing valid on arbitrary convex normal neighborhoods of time oriented Lorentzian manifolds. This definition is a function defined at any two points along a null geodesic that lie in a convex normal neighborhood, and foregoes the usual notions of lens and source planes in gravitational lensing. Rather, it makes essential use of the van Vleck determinant, which we present via the exponential map, and Etherington's definition of luminosity distance for arbitrary spacetimes. We then specialize our definition to spacetimes, like Schwarzschild's, in which the lens is compact and isolated, and show that our magnification function is monotonically increasing along any geodesic contained within a convex normal neighborhood.

gr-qc↗

arXiv2015

A geometrical approach to gravitational lensing magnification

The standard definition of gravitational lensing magnification is generalized to Lorentzian spacetimes, and it is shown how it can be interpreted geometrically in terms of the van Vleck determinant and the exponential map. This is joint work with Amir B. Aazami (Kavli IPMU, University of Tokyo).

gr-qc↗

arXiv2014

Detection of the Gravitational Lens Magnifying a Type Ia Supernova

Objects of known brightness, like Type Ia supernovae (SNIa), can be used to measure distances. If a massive object warps spacetime to form multiple images of a background SNIa, a direct test of cosmic expansion is also possible. However, these lensing events must first be distinguished from other rare phenomena. Recently, a supernova was found to shine much brighter than normal for its distance, which resulted in a debate: was it a new type of superluminous supernova or a normal SNIa magnified by a hidden gravitational lens? Here we report that a spectrum obtained after the supernova faded away shows the presence of a foreground galaxy--the first found to strongly magnify a SNIa. We discuss how more lensed SNIa may be found than previously predicted.

astro-ph.CO↗

arXiv2013

On de-Sitter Geometry in Cosmic Void Statistics

Starting from the geometrical concept of a 4-dimensional de-Sitter configuration of spheres in Euclidean 3-space and modelling voids in the Universe as spheres, we show that a uniform distribution over this configuration space implies a power-law for the void number density which is consistent with results from the excursion set formalism and with data, for an intermediate range of void volumes. The scaling dimension of the large scale structure can be estimated as well. We also discuss the effect of restricting the survey geometry on the void statistics. This work is a new application of de-Sitter geometry to cosmology and also provides a new geometrical perspective on self-similarity in cosmology.

astro-ph.CO↗

arXiv2013

Extraordinary Magnification of the Ordinary Type Ia Supernova PS1-10afx

Recently, Chornock and co-workers announced the Pan-STARRS discovery of a transient source reaching an apparent peak luminosity of ~4x10^44 erg s^-1. We show that the spectra of this transient source are well fit by normal Type Ia supernova (SNIa) templates. The multi-band colors and light-curve shapes are also consistent with normal SNeIa at the spectroscopically determined redshift of z=1.3883; however, the observed flux is a constant factor of ~30 times too bright in each band over time as compared to the templates. At minimum, this shows that the peak luminosities inferred from the light-curve widths of some SNeIa will deviate significantly from the established, empirical relation used by cosmologists. We argue on physical grounds that the observed fluxes do not reflect an intrinsically luminous SNIa, but rather PS1-10afx is a normal SNIa whose flux has been magnified by an external source. The only known astrophysical source capable of such magnification is a gravitational lens. Given the lack of obvious lens candidates, such as galaxy clusters, in the vicinity, we further argue that the lens is a supermassive black hole or a comparatively low-mass dark matter halo. In this case, the lens continues to magnify the underlying host galaxy light. If confirmed, this discovery could impact a broad range of topics including cosmology, gamma-ray bursts, and dark matter halos.

astro-ph.CO↗

arXiv2012

On de-Sitter geometry in crater statistics

The cumulative size-frequency distributions of impact craters on planetary bodies in the solar system appear to approximate a universal inverse square power-law for small crater radii. In this article, we show how this distribution can be understood easily in terms of geometrical statistics, using a de-Sitter geometry of the configuration space of circles on the Euclidean plane and on the unit sphere. The effect of crater overlap is also considered.

astro-ph.EP↗

arXiv2007

A Lefschetz fixed point theorem in gravitational lensing

Topological invariants play an important rôle in the theory of gravitational lensing by constraining the image number. Furthermore, it is known that, for certain lens models, the image magnifications $μ_i$ obey invariants of the form $\sum_i μ_i=1$. In this paper, we show that this magnification invariant is the holomorphic Lefschetz number of a suitably defined complexified lensing map, and hence a topological invariant. We also provide a heat kernel proof of the holomorphic Lefschetz fixed point formula which is central to this argument, based on Kotake's proof of the more general Atiyah-Bott theorem. Finally, we present a new astronomically motivated lens model for which this invariant holds.

math-ph↗